# “Trigonometry’s beginning”

**By Scott Hamilton**

If you ever took a course in trigonometry, there is one very famous name you are sure to remember, Pythagoras. The main theory that makes all of trigonometry work was supposedly first discovered by Pythagoras between 540-495 BC. It states that the area of a square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides: written mathematically as a^{2} + b^{2} = c^{2}. However, he was not the first to make this discovery, nor was he the only mathematician to prove the theory.

I guess first I should define a few of the terms in his theory for those that don’t know anything about trigonometry. The first term, of course, is trigonometry itself, which is simply the study of triangles. A triangle is a two-dimensional shape with exactly three straight sides and three angles. Pythagoras’ theorem is about a right triangle, where one of the three angles is exactly 90 degrees. If you don’t know what 90 degrees means, it is the angle at which any two sides of a square meet, so a right-triangle can be formed by cutting the corner off a square with a single cut at any angle.

What Pythagoras noticed was that if you put perfect squares on each side of a right triangle, the squares of the two smaller sides had the same area as the square that made up the longest side. The longest side of a right-triangle is always the side opposite from the right angle and has a special name, called the hypotenuse. If you want to see the proofs of his theorem, Wikipedia has a great write-up on it at https://en.wikipedia.org/wiki/Pythagorean_theorem.

The Pythagorean theorem resulted in a table of numbers called the Pythagorean triples, which are the list of numbers corresponding to common right triangle sizes. Some of the most well known are (3,4,5) and (5, 12, 13); in fact, most roofs in the northern United States are built on what we call a 5/12 pitch, which means that each side of the roof truss is a right triangle with the (5, 12, 13) Pythagorean triple as its main ratio. The roof rises five inches for every 12 inches at the base and needs a board 13 inches long to accomplish the goal.

It was up until just a few years ago that mathematicians believed Pythagoras was the first to make this discovery. However, a recent translation of a 3,700-year-old Babylonian tablet proves that the Babylonians made the exact same discovery more than a thousand years before. The etchings on the tablet dated back to the Old Babylonian Period between 1900 and 1600 B.C. and was originally thought to be a land survey map. However, a more recent translation of the tablet discovered that the land surveyors were using “Pythagorean triples” to make the precise right angles on the survey plats. Not only were they Pythagorean triples used directly on the map, but the tablet also contained a nearly complete table of the triples for quick reference by the surveyors. Of course, the table is not labeled Pythagorean triples because it was carved over 1,000 years before Pythagoras was born.

It was originally thought that trigonometry was born in ancient Greece to assist with measuring and studying the night sky. However, this new discovery shows that it was birthed by Babylon to measure the ground and not the sky. To me it just goes to prove that we were a lot smarter a lot sooner than history tells us. Until next week stay safe and learn something new.

*Scott Hamilton is an Expert in Emerging Technologies at ATOS and can be reached with questions and comments via email to sh*******@te**********.org or through his website at https://www.techshepherd.org.*